1994/02/21 by Brian R. Greene, Brian Greene, David R. Morrison +1 · 3 citations
Mathematics · Physics and Astronomy · #Geometric and Algebraic Topology #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #alg-geom #hep-th #math.AG
paper · pdf · doi:10.1007/bf02101657
published as Commun.Math.Phys.173:559-598,1995 · 44 pages plus 3 tables using harvmac
arxiv created 1994/02/21 · openalex publication_date 1995/11/01 · arxiv updated 2011/10/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We describe mirror manifolds in dimensions different from the familiar case of complex threefolds. We emphasize the simplifying features of dimension three and supply more robust methods that do not rely on such special characteristics and hence naturally generalize to other dimensions. The moduli spaces for Calabi--Yau d-folds are somewhat different from the ``special Kähler manifolds'' which had occurred for d=3, and we indicate the new geometrical structures which arise. We formulate and apply procedures which allow for the construction of mirror maps and the calculation of order-by-order instanton corrections to Yukawa couplings. Mathematically, these corrections are expected to correspond to calculating Chern classes of various parameter spaces (Hilbert schemes) for rational curves on Calabi--Yau manifolds. Our results agree with those obtained by more traditional mathematical methods in the limited number of cases for which the latter analysis can be carried out. Finally, we make explicit some striking relations between instanton corrections for various Yukawa couplings, derived from the associativity of the operator product algebra.